Journal of Computational Physics 243 (2013) 323–343

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Journal of Computational Physics

journal homepage: www.elsevier .com/locate / jcp

Exact PDF equations and closure approximationsfor advective-reactive transport

0021-9991/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jcp.2013.03.001

⇑ Corresponding author. Tel.: +1 401 863 1217.E-mail addresses: daniele_venturi@brown.edu, venturi.daniele@gmail.com (D. Venturi), george_karniadakis@brown.edu (G.E. Karniadakis).

D. Venturi a, D.M. Tartakovsky b, A.M. Tartakovsky c, G.E. Karniadakis a,⇑a Division of Applied Mathematics, Brown University, Providence, RI 02912, USAb Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USAc Pacific Northwest National Laboratory, Richland, WA 99352, USA

a r t i c l e i n f o

Article history:Received 28 March 2012Accepted 5 March 2013Available online 15 March 2013

Keywords:PDF methodsUncertainty quantificationHeterogeneous reactionStochastic modelingGeochemistry

a b s t r a c t

Mathematical models of advection–reaction phenomena rely on advective flow velocityand (bio) chemical reaction rates that are notoriously random. By using functional integralmethods, we derive exact evolution equations for the probability density function (PDF) ofthe state variables of the advection–reaction system in the presence of random transportvelocity and random reaction rates with rather arbitrary distributions. These PDF equationsare solved analytically for transport with deterministic flow velocity and a linear reactionrate represented mathematically by a heterogeneous and strongly-correlated random field.Our analytical solution is then used to investigate the accuracy and robustness of therecently proposed large-eddy diffusivity (LED) closure approximation [1]. We find thatthe solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates,is accurate even in the presence of strong correlations and it provides an upper bound ofpredictive uncertainty.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The mathematical modeling of realistic advection–reaction phenomena involve many random parameters. For example,the dynamics of solute transport in an heterogeneous porous medium is significantly influenced by the epistemic uncer-tainty in the porosity distribution, advective (Darcy) flow velocity and (bio) chemical reaction rates [2]. These random fieldsrender the governing equations of the advection–reaction system stochastic. A common approach to quantifying the statis-tical properties of the state variables of the system, e.g., the solute concentration, relies in modeling their probability densityfunctions (PDFs) directly via deterministic equations. This has advantages over other uncertainty quantification methods,such as polynomial chaos [3–6], probabilistic collocation [7,8], perturbation methods [35–38], and generalized spectraldecompositions [9–13]. In particular, PDF methods allow to directly ascertain the tails of probabilistic distributions thusfacilitating the assessment of rare events and associated risks. This is in contrast to many other stochastic approaches thatuse the variance (or the standard deviation) of a system variable as a measure of predictive uncertainty. Another key advan-tage of PDF methods is that they do not suffer from the curse of dimensionality. Moreover, they can be used to tackle severalopen problems in stochastic dynamics such as discontinuities in parametric space [6] and long-term integration [14,15].

324 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

As is well known, in general it is not possible to determine an exact one-point (in space and time) PDF equation for thesolution to an arbitrary partial differential equation. This is due to the fact that the solution to the PDE can be nonlocal inspace and time.1 However, a functional differential equation for the Hopf characteristic functional can always be determined[17–19], although effective analytical or numerical methods to solve it are still lacking. Moreover, the Hopf functional equationis equivalent to an infinite hierarchy of equations involving multi-point PDFs of increasing order [20–23].

First-order stochastic partial differential equations (SPDEs), such as the advection–reaction equation, are often amenableto exact treatment with PDF methods [1,24–26]. The main reason is that these equations can be reduced to a finite set ofordinary differential equations along the characteristic curves. This enables one to obtain an exact evolution equation forthe joint PDF of the state variables and the external random fields appearing in the SPDE. In general, these random fieldsare high-dimensional, e.g., represented in terms of many random variables in a Karhunen–Loève expansion, and thereforethe corresponding exact equation for the joint PDF is high-dimensional as well [27,26]. From a computational viewpoint thiscould be an issue despite the recent advances in numerical methods for high-dimensional problems such as proper general-ized decomposition [28–30], sparse grid collocation [31,7] or functional ANOVA techniques [32–34].

A closure approximation can significantly reduce the number of parameters appearing in the PDF equation and thereforeit can provide an effective computational tool that allows for an efficient integration. In particular, the large-eddy diffusivity(LED) closure for advection–reaction equations [1] has been shown to be effective for uncorrelated and weakly correlatedrandom reaction rates. However, the performance of the LED-based PDF equations for strongly correlated reaction rates re-mains unexplored. Its investigation is the main objective of the present study. To this end, we consider the prototype prob-lem proposed in [1] and obtain analytical solutions to the PDF equation for two different random reaction models. Theanalytical solutions will be employed as useful benchmark to test the accuracy and effectiveness of the LED closure.

This paper is organized as follows. In Section 2 we formulate the governing equations of advective-reactive transport inporous media and obtain an evolution law for the corresponding indicator function ([39, Ch. 3]). Section 2.2 presents theLED-based PDF equation for random advection–reaction transport. The exact evolution equation for the joint response-exci-tation PDF of the advection–reaction system is derived and discussed in Section 2.3. In Section 3 and Section 4 we comparethe LED approximation with exact analytical results for a prototype advection–reaction problem involving linear reactionsand strongly correlated random reaction rates. Finally, the main findings and their implications are summarized in Section 5.We also include two brief appendices, where we obtain analytical solutions to the equation for the joint response-excitationPDF and the advection–reaction equation in physical space.

2. Problem formulation

Let us consider the dimensionless form of the advection–reaction equation for a scalar concentration field cðx; tÞ

1 Percase we

wherepropertall spat

2 Werescalinstochas

3 It isdifferen

@c@t¼ �r � ðucÞ þ DafjðcÞ; f jðcÞ ¼ �ajðca � 1Þ; ð1Þ

where uðxÞ is the dimensionless advective velocity; jðxÞ and a are the dimensionless reaction rate and the stoichiometriccoefficient, respectively; and Da is the dimensionless Damköhler number.2 The source function fjðcÞ provides a macroscopic(continuum-scale) representation of a heterogeneous precipitation/dissolution reaction.3 Eq. (1) is subject to suitable boundaryconditions and the initial condition

cðx;0Þ ¼ C0ðxÞ; ð2Þ

where C0ðxÞ denotes the initial concentration.Spatial heterogeneity and data scarcity render both advective velocity uðxÞ and reaction rate jðxÞ uncertain. To quantify

the impact of this parametric uncertainty on predictions of concentration cðx; tÞ, we treat these quantities as random fields,i.e., we consider uðx;xÞ and jðx;xÞ, with x indicating an element of the sample space in a suitable probability space. Addi-tionally, we account for uncertainty in the initial concentration, i.e. we consider C0ðx;xÞ. Available data or expert opinion(see, e.g., [24,25,1]) can be used to statistically characterize the random reaction rate jðx;xÞ; in this paper we will discusstwo different models defined in terms of chi-squared and uniform random variables (see Section 3.1). Statistics of the

haps one of the simplest example is the solution to heat equation in a one-dimensional unbounded spatial domain for a random initial condition. In thishave the random field [16]

hðx; t;xÞ ¼Z 1

�1Gðx; x0 jt;0Þh0ðx0; xÞdx0

G denotes the Green function of the one-dimensional heat equation and h0ðx;xÞ is the random initial temperature condition. Clearly, the statisticalies at location x and time t þ dt are influenced by the joint probability functional of hðx; t; xÞ at time t, i.e. the joint probability density of hðx; t; xÞ atial points. Thus, no closed one-point (in space and time) PDF equation exists for the solution to the heat equation.recall that Da ¼def DrjCa�1

eq =V , where D denotes the reference length, V the reference velocity, Ceq the deterministic equilibrium concentration, rj a scalarg factor for the reaction rate j and a the stoichiometric coefficient. The quantity rj can be, e.g., the mean reaction rate if this is a uniform (in space)tic process.

worthwhile emphasizing that the methodology described in this paper is also applicable to other types of chemical reactions, defined in terms oft fjðcÞ.

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 325

macroscopic flow velocity uðx; xÞ, including its spatially varying mean huðx;xÞi and correlation huiðx;xÞujðy;xÞi(i; j ¼ 1;2;3), can be determined by solving the flow equations with random hydraulic conductivity or random initial andboundary conditions (e.g., [40,41]).

2.1. Equations for the indicator function

Let us consider the function

Pða; x; t;xÞ � d a� cðx; t;xÞð Þ; ð3Þ

where dð�Þ is the Dirac delta and a is a deterministic value that the random concentration c can take at a space-time pointðx; tÞ. The ensemble average (over random c) of P is the one-point (in space and time) PDF of the concentration cðx; t;xÞ,

pðaÞcðx;tÞ ¼defhPða; x; t;xÞi: ð4Þ

By using elementary properties of the Dirac delta function [42], it is easy to show that Pða; x; t;xÞ satisfies a linear advectionequation (see [39, Ch. 3])

@P@tþr � ðuPÞ ¼ �Da

@ fjðaÞP½ �@a

: ð5Þ

Following [1], we introduce a four-dimensional space with coordinates ~x ¼ ðx1; x2; x3; x4 � aÞ, in which the gradient operatorand the velocity field are defined as

~r ¼def @

@x1;@

@x2;@

@x3;@

@x4� @

@a

� �; ~u ¼defðu1;u2;u3;u4 � DafjðaÞÞ: ð6Þ

This allows us to rewrite (5) in the compact form

@P@tþ ~r � ð~uPÞ ¼ 0: ð7Þ

Stochastic averaging of this equation yields a one-point PDF equation

@pðaÞcðx;tÞ

@t¼ � ~r � h~uipðaÞcðx;tÞ

h i� ~r � h~u0Pi; ð8Þ

where the prime denotes the fluctuating part of the Reynolds decomposition of ~u. The cross-correlation term h~u0Pi in (8)requires a closure approximation. Over the years, and in different contexts, many closures (e.g., [43–45]) have been proposedin order to express h~u0Pi in terms of the one-point PDF pðaÞcðx;tÞ. One of these methods is described below.

2.2. Large-eddy diffusivity (LED) approximation

The Large-eddy diffusivity (LED) approximation [1] is a phenomenological closure that allows one to replace the PDF Eq.(8) with a Fokker–Planck type equation

@pðaÞcðx;tÞ

@t¼ �

@ UipðaÞcðx;tÞ

h i@~xi

þ @

@~xjDij

@pðaÞcðx;tÞ

@~xi

" #; i; j ¼ 1; . . . ;4: ð9Þ

The ‘‘effective diffusivity’’ Dij and the ‘‘effective velocity’’ U are defined as

Dijð~x; tÞ¼defZ t

0

Z~Xh~u0ið~xÞ~u0jð~yÞGð~x; ~y; t � sÞid~yds; i; j ¼ 1; . . . ;4; ð10Þ

Uð~x; tÞ¼defh~ui � DaZ t

0

Z~Xh~u0ð~xÞ @f 0jð~yÞ

@y4Gð~x; ~y; t � sÞid~yds; ð11Þ

where ~u0jð~yÞ denotes the velocity fluctuation arising from the Reynolds decomposition of the four-dimensional random veloc-ity as ~u ¼ h~ui þ ~u0, and G is the random Green function associated with (7). The latter is defined as a solution of

@G@sþ ~u � ~rG ¼ �dð~x� ~yÞdðt � sÞ; ð12Þ

subject to the appropriate homogeneous initial and boundary conditions. To render the diffusion tensor (10) and effectivevelocity (11) computable, we approximate the third-moments in their integrands as follows

Dijð~x; tÞ ’Z t

0

Z~Xh~u0ið~xÞ~u0jð~yÞiGð~x; ~y; t � sÞd~yds; ð13Þ

326 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

Uð~x; tÞ ’ h~ui � DaZ t

0

Z~Xh~u0ð~xÞj0ð~yÞi @faðy4Þ

@y4Gð~x; ~y; t � sÞd~yds: ð14Þ

Here faðy4Þ ¼def�aðya

4 � 1Þ and G is a deterministic Green function defined as a solution to the deterministic advectionequation

@G@sþ h~ui � ~rG ¼ �dð~x� ~yÞdðt � sÞ: ð15Þ

This approximation has been shown to be effective if pðaÞcðx;tÞ varies slowly in time and space relatively to ~u.

2.3. Exact joint PDF equation

Let us assume that the random coefficients jðx;xÞ and uðx;xÞ can be represented, with a given degree of accuracy, withfinite series

jðx; xÞ ¼XN

j¼1

vjðxÞ/jðxÞ; uðx;xÞ ¼XM

j¼1

gjðxÞujðxÞ; ð16Þ

where vðxÞ ¼ fv1; . . . ;vNg and gðxÞ ¼ fg1; . . . ;gMg are two sets of random variables with known joint probability densityfunction. Then, in analogy with (4), the joint PDF of the concentration field cðx; t;xÞ and the random vectors vðxÞ andgðxÞ can be expressed as (e.g., [20])

pða;b;zÞcðx;tÞvg ¼ hdða� cðx; t;xÞÞYN

j¼1

dðbj � vjðxÞÞYMk¼1

dðzk � gkðxÞÞi; ð17Þ

where the elements of sets b ¼ fb1; . . . ; bNg and z ¼ fz1; . . . ; zMg represent deterministic values that the corresponding ran-dom elements of the sets vðxÞ and gðxÞ can take on, and the stochastic average is with respect to the joint probability func-tional of the variables vðxÞ and gðxÞ and all the other input processes such as random boundary and initial conditions.Differentiating (17) with respect to time t and making use of (1) yields

@P@t¼ � @

@ahdða� cÞ @c

@t

YNj¼1

dðbj � vjÞYMk¼1

dðzk � gkÞi

¼ � @

@ahdða� cÞ �r � ðucÞ þ DafjðcÞ½ � @c

@t

YNj¼1

dðbj � vjÞYMk¼1

dðzk � gkÞi; ð18Þ

where P is a shorthand notation for the multi-dimensional joint PDF (17). Substituting (16) into (18), we find that P satisfies alinear advection equation in the variables (t; x; a) with ðN þMÞ parameters,

@P@t¼ @

@aaPXM

j¼1

zjr �uj

!�rP �

XM

j¼1

zjuj

!� Da

@

@aaa � 1ð ÞP

XN

j¼1

bj/j

!: ð19Þ

The one-point PDF of the concentration c is obtained by marginalizing P with respect to all the parameters b and z,

pðaÞcðx;tÞ ¼Z 1

�1� � �Z 1

�1pða;b;zÞcðx;tÞvgdb1 . . . dbNdz1 . . . dzM : ð20Þ

The corresponding (integro-differential) evolution equation for pðaÞcðx;tÞ can be obtained by integrating (19) with respect tothe variables b and z.

The following points are worthwhile emphasizing. First, apart from the decomposition of the random coefficients u and jinto the finite-term series (16), the evolution equation for the joint PDF (19) is exact, i.e, it requires no closure approxima-tions. Second, the parametric space of the joint PDF (19) is significantly higher than that of the LED-based PDF Eq. (9). Third,the sequence of steps leading to the PDF Eqs. (9) and (19) can be applied to quantify parametric uncertainty in more generalnonlinear first-order PDEs of the form @c=@t þ Pðc; t; x;vÞ � rxc ¼ Qðc; t; x;gÞ [26,46].

3. Comparison between LED-based and exact PDF equations

To elucidate the salient features of the two approaches to derivation of PDF equations, we consider a one-dimensionaladvection-reaction Eq. (1) with a linear reaction term (a ¼ 1). Mass conservation requires the dimensionless advection veloc-ity u to be constant and deterministic (e.g., [1], §4). Without loss of generality, we set u ¼ 1 so that the reaction rate jðxÞ isthe only source of uncertainty. Under these conditions, (1) reduces to

@c@t¼ � @c

@x� Da jðx;xÞ c � 1ð Þ x 2 ½0; L�: ð21Þ

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 327

We supplement (21) with the following boundary and initial conditions

4 Thethey incformula

cð0; t; xÞ ¼ 0; cðx; 0;xÞ ¼ 0: ð22Þ

To ensure the well-posedness of the initial-boundary value problem (21), (22), the characteristic curves [47] of (21) arisingfrom the initial-boundary data should not intersect each others. In other words, the characteristic curves originating fromt ¼ 0 should not intersect the ones from x ¼ 0 (see Section 3.4 for more details).

3.1. Modeling of the random reaction rate

The previous analyzes (e.g., [24,25]) treated the random reaction rate jðx;xÞ as a statistically homogeneous multi-variatelog-normal field. In this paper we will employ two alternative statistical models of jðx;xÞ, both of which are describedbelow.

3.1.1. Reaction rate model based on chi-squared random variablesLet us represent the random reaction rate jðx;xÞ on the one-dimensional interval ½0; L� with a finite-term series

jðx; xÞ ¼XN

j¼1

vjðxÞ/jðxÞ; x 2 ½0; L�: ð23Þ

The modes /kðxÞ and the random variables vkðxÞ have to be selected in a way that ensures that jðx;xÞ remains positive withprobability one. We achieve this requirement by taking vkðxÞ to be independent chi-squared random variables. This yields arandom reaction rate whose one-point distribution function can be represented in terms of an infinite series of incompletegamma integrals [48] or other series involving normal variables [49–52]. Among several possible choices, we consider therepresentation given by an infinite linear combination of chi-squared distribution functions with an arbitrary scale param-eter b > 0 (e.g., Theorem 2 in [49], or §6 in [50]). Let us denote the distribution function of jðx;xÞ by

PðaÞjðxÞ ¼defPr

XN

j¼1

vjðxÞ/jðxÞ 6 a

" #: ð24Þ

For all x such that /jðxÞ– 0 (j ¼ 1; . . . ;N),4 the (central) chi-squared distribution function with N degrees of freedom is given by

PðaÞjðxÞ ¼X1j¼0

cjðxÞFNþ2jab

� �; FnðaÞ ¼

1

2n=2Cðn=2Þ

Z a

0e�y=2yn=2�1dy; ð25Þ

where the coefficients cjðxÞ satisfy the recursion relationship

cjðxÞ ¼12j

Xj�1

r¼0

gj�rðxÞcrðxÞ; c0 ¼YNi¼1

b/iðxÞ

� �1=2

; gmðxÞ ¼XN

i¼1

1� b/iðxÞ

� �m

: ð26Þ

The series (25) converges uniformly in every finite interval of a [49]. Differentiating the distribution function (25) with re-spect to a, we obtain the one-point PDF of the random reaction rate (23),

pðaÞjðxÞ ¼1b

X1j¼0

cjðxÞqNþ2jab

� �; where qnðsÞ ¼

e�s=2sn=2�1

2n=2Cðn=2Þ: ð27Þ

The correlation structure of the random field (23) is obtained as

hjðx;xÞjðy; xÞi ¼ 3XN

i¼1

/iðxÞ/iðyÞ þXN

i;j¼1i–j

/iðxÞ/jðyÞ; ð28Þ

while the covariance function is

Cjðx; yÞ¼defhjðx; xÞjðy; xÞi � hjðx;xÞihjðy;xÞi ¼ 2XN

i¼1

/iðxÞ/iðyÞ: ð29Þ

Among many possible choices of the bases functions /jðxÞ, in this paper we consider

/jðxÞ ¼1kj

sinjL

x� �2

; k > 1: ð30Þ

results presented in [49] are valid in the more general case of arbitrary non-negative quadratic forms of standard Gaussian variables. This means thatlude also the cases where some of the /i in (23) are zero. In this circumstance we can simply remove those terms from the series (23) and then apply(25).

328 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

The corresponding random samples of jðx;xÞ and its covariance are shown in Fig. 1 for several k. Note that the random reac-tion rate is positive with probability 1 (as it should be), and at x ¼ 0 it is always zero by construction. Fig. 2 exhibits the rel-ative L1 norm of each term in the series expansion (23). This quantity is defined as

Fig. 1.(30)).

lk¼def ekX

j

ej

; ek¼defZ L

0j/kðxÞjdx: ð31Þ

It represents the relative contribution of each term in the representation of the random field jðx;xÞ as a function of theparameter k. Substituting 30 into (31) yields

ek ¼2k� sinð2kÞ

4kkk=L: ð32Þ

Let

N ¼maxk

lk

� �P s ð33Þ

define the dimensionality of the random field jðx;xÞ, i.e., the number of random variables in the series (23) that is requiredto reach a certain threshold s. The table in Fig. 2 shows how the dimensionality N increases with the decreasing values of k,

Covariance function (28) (left column) and random samples of the reaction rate (23) (right column) for different values of the parameter k (see Eq.

Fig. 3. (a) One-point probability density function (27) of the random reaction rate jðx;xÞ for k ¼ 1:5. (b) Mean and standard deviation of jðx;xÞ.

Fig. 2. Relative L1 norm (31) of each term in the series representation of the random reaction rate (23). Shown are results for different k (see Eq. (30)). Thedimensionality of the random reaction rate, i.e., number of random variables in Eq. (23), is shown in the table for a threshold set at s ¼ 0:05%.

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 329

for the threshold s ¼ 0:05%. Fig. 3 provides an example of the one-point PDF (27), the mean, and standard deviation ofjðx;xÞ corresponding to k ¼ 1:5.

3.1.2. Reaction rate model based on uniform random variablesThe second finite-series representation the random reaction rate jðx;xÞ is based on a truncated Karhunen–Loève

expansion

5 Not

jðx; xÞ ¼ K þffiffiffi3p

rjffiffiffiffiffi2‘p

XN

j¼1

ffiffiffiffihj

pfjðxÞgjðxÞ; ð34Þ

where K is a positive constant, and fjðxÞ are zero-mean independent uniform random variables in ½�1;1�. The quantities hj

and gjðxÞ denote, respectively, the eigenvalues and the eigenfunctions of the exponential covariance function

Cjðx; yÞ ¼r2

j2‘

e�jx�yj=‘; ð35Þ

where ‘ is the correlation length.5 The factorffiffiffi3p

on the right-hand-side of (34) reflects the choice of hf2j i ¼ 1=3 and ensures that

(34) and (35) are consistent. The eigenvalues hj and eigenfunctions gjðxÞ of the exponential correlation function (35) are given by(e.g., [53])

hj ¼2‘

‘2z2j þ 1

; gjðxÞ ¼zj‘

AjcosðzjxÞ þ

1Aj

sinðzjxÞ; ð36Þ

where zj are zeros of the transcendental equation

z2i �

1‘2

� �tanðziLÞ � 2

zi

‘¼ 0 ð37Þ

and

e that Cjðx; yÞ is an element of a delta sequence that converges to r2jdðx � yÞ as ‘! 0.

Fig. 4.to be e

Fig. 5.Same aand ind

330 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

Aj ¼def

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

1þ z2j ‘

2

2þ

z2j ‘

2 � 14zj

sinð2zjLÞ þ ‘1� cosð2zjLÞ

2

s: ð38Þ

In analogy with the previous section, we define the relative spectrum Hj and the dimensionality N of the Karhunen–Loèveexpansion (34) for a given threshold level s as

Hj ¼hjX

k

hk

; N ¼maxi½Hi� > s: ð39Þ

Fig. 4 shows Hj and N for s ¼ 0:05% and different correlation lengths ‘. For each correlation length ‘, the constant K in (34)can be selected to ensure that jðx;xÞ is positive with probability one (Fig. 5). The lower bound of K

ffiffiffiffiffi2‘p

=ðffiffiffi3p

rjÞ in Fig. 5 isvalid for uniform variables in ½�1;1�. It is obtained numerically by first sampling (34) and then computing the minimum overall the realizations. For other types of random variables the bounds in Fig. 5 are different.Several realization of the random reaction rate for different correlation lengths and perturbation amplitude rj

ffiffiffi3p

=ffiffiffiffiffi2‘p

¼ 0:25are shown in Fig. 6. The value of K in (34) is selected in order to satisfy the positivity condition. The one-point PDF of the Karh-unen–Loève series (34) with uniformly distributed fi has a nontrivial mathematical expression [54,55]; it is plotted in Fig. 7 fordifferent correlation lengths ‘. Note that as ‘! 0 (i.e. for N !1), the PDF becomes Gaussian, in agreement with the centrallimit theorem. In other words, the random reaction rate (34) becomes Gaussian white noise as ‘! 0.

3.2. Exact joint PDF equation

The joint PDF equation for the random concentration cðx; t;xÞwhose dynamics is governed by (21) with random reactionrate (23) is (see Section 2.3)

@P@t¼ � @P

@x� Da

@½ð1� aÞP�@a

XN

i¼1

/iðxÞbi; ð40Þ

Spectral decay of the Karhunen–Loève expansion (34) as a function of the correlation length ‘. The covariance of the random reaction rate is assumedxponential. The table on the right shows the number of expansion terms in (34) for L ¼ 200 and an energy threshold set at s ¼ 0:05%.

(a) Lowest values of the constant Kffiffiffiffiffi2‘p

=ðffiffiffi3p

rjÞ for which the exponentially correlated random reaction rate (34) is positive with probability one. (b)s figure (a) but here we show rj

ffiffiffi3p

=ðKffiffiffiffiffi2‘pÞ versus ‘. These plots are valid when the random variables fkðxÞ appearing in (34) are uniform in ½�1;1�

ependent to each other.

Fig. 6. Samples of the exponentially correlated random reaction rate (34) for K ¼ 0:75 and rjffiffiffi3p

=ffiffiffiffiffi2‘p

¼ 0:25. We show results corresponding to differentcorrelation lengths. Note that with this choice of parameters the the positivity condition of Fig. 5 is satisfied for all ‘ > 1 since K

ffiffiffiffiffi2‘p

=ðffiffiffi3p

rjÞ ¼ 3.

Fig. 7. One-point PDF of the random reaction rate (34) for rjffiffiffi3p

=ffiffiffiffiffi2‘p

¼ 0:25, K ¼ 0:75, and different correlation lengths ‘.

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 331

where P is a shorthand notation for

6 Somparame

and the

Once th

pða;bÞcðx;tÞv ¼ hdða� cðx; t;xÞÞYN

k¼1

dðbk � vkðxÞÞi: ð41Þ

Integration of (40) with respect to b ¼ fb1; . . . ; bNg yields an evolution equation for the one-point PDF of the random concen-tration field,

@pðaÞcðx;tÞ

@t¼ �

@pðaÞcðx;tÞ

@x� Da

@

@að1� aÞ

XN

i¼1

/iðxÞZ 1

�1bip

ða;biÞcðx;tÞvi

dbi

" #: ð42Þ

This equation remains unclosed since the dynamics of the one-point PDF pðaÞcðx;tÞ depends on N joint PDFs pða;biÞcðx;tÞvi

. Auxiliary con-ditions for the joint PDF Eq. (40) are derived from the initial and boundary conditions (22). Taking the deterministic concen-tration outside the averaging operator in the definition (41), we obtain the boundary and initial conditions6

pða;bÞcð0;tÞv ¼ dða� 0ÞpðbÞv ; pða;bÞcðx;0Þv ¼ dða� 0ÞpðbÞv : ð45Þ

Assignment of boundary conditions with respect to a is nontrivial. If the joint PDF pða;bÞcðx;tÞv were defined on the interval�1 < a <1, then a natural boundary condition to use would be pð�1;bÞcðx;tÞv ¼ 0. However, it follows from the problem formu-lation (21), (22) that cðx; t;xÞ 2 ½0;1� with probability 1. This, in turn, implies that both the joint PDF Eq. (40) and its single-point counterpart (42) are defined on the compact support a 2 ½0;1� for which PDF boundary conditions are not clearly de-fined. In the following section, we remove this ambiguity by reformulating the problem in terms of cumulative density

etimes it may not be convenient to have a Dirac delta function at the boundary or at the initial condition of a PDE. In these circumstances one can firsttrize such conditions by using, e.g., an element of a delta sequence ([42, p. 58]) D�ðaÞ, e.g.,

D�ðaÞ ¼1

Cðq=2Þ2q=2�a�

� �q=2e�a=ð2�Þ ðq 2 NÞ; lim

�!0D�ðaÞ ¼ dðaÞ; ð43Þ

n compute the solution to the PDF equation corresponding to the regularized boundary and initial conditions

pða;b1 ;���;bN Þcð0;tÞv1 ���vN

¼ D�ðaÞpðb1 ;���;bN Þv1 ���vN

; pða;b1 ;���;bN Þcðx;0Þv1 ���vN

¼ D�ðaÞpðb1 ;���;bN Þv1 ���vN

: ð44Þ

e solution to the regularized problem is available, then we set �! 0 to obtain the desired solution.

332 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

functions instead of PDFs. In appendix A, we use the method of characteristics (e.g., [47]) to analytically solve the joint PDFEq. (40) subject to the auxiliary conditions (45).

3.3. LED-based PDF equation

For the problem under consideration, the general PDF Eq. (9) reduces to

@pðaÞcðx;tÞ

@t¼ �

@pðaÞcðx;tÞ

@x�@ U4pðaÞcðx;tÞ

h i@a

þ @

@aD44

@pðaÞcðx;tÞ

@a

" #; ð46Þ

where the diffusion coefficient and the effective velocity are given by [1]

D44ðx; a; tÞ ¼ Da2ða� 1Þ2R t

0 Cjðx; x� bÞe2bDadb t 6 TðaÞR T0 Cjðx; x� bÞe2bDadb t > TðaÞ

(ð47Þ

and

U4ðx; a; tÞ ¼ hjðx;xÞiDað1� aÞ � Da2ða� 1ÞR t

0 Cjðx; x� bÞebDadb t 6 TðaÞR T0 Cjðx; x� bÞebDadb t > TðaÞ

(; ð48Þ

with

TðaÞ ¼def 1Da

ln1

1� a

� �: ð49Þ

Eq. (46), together with the boundary and initial conditions

pðaÞcðx;0Þ ¼ dðaÞ; pðaÞcð0;tÞ ¼ dðaÞ; ð50Þ

can be rewritten in terms of the cumulative distribution function

PðaÞcðx;tÞ ¼Z a

0pða

0 Þcðx;tÞda0: ð51Þ

The result is

@PðaÞcðx;tÞ

@t¼ �

@PðaÞcðx;tÞ

@x� U4ðx; a; tÞ

@PðaÞcðx;tÞ

@aþ D44ðx; a; tÞ

@2PðaÞcðx;tÞ

@a2 þ Jðx; a; tÞja¼0; ð52Þ

where

Jðx; a; tÞ ¼def U4ðx; a; tÞ@PðaÞcðx;tÞ

@a� D44ðx; a; tÞ

@2PðaÞcðx;tÞ

@a2 ð53Þ

denotes the probability current. Eq. (52) is second-order in a and first-order in x and t. Therefore, it requires two additionalconditions in a, namely,

Pð0Þcðx;tÞ ¼ 0; Pð1Þcðx;tÞ ¼ 1; ð54Þ

one condition in x

PðaÞcð0;tÞ ¼ 1 ð55Þ

and the initial condition

PðaÞcðx;0Þ ¼ 1: ð56Þ

Both the effective velocity and the diffusion tensor decrease as the Damköhler number Da becomes smaller. In the limitDa! 0;U4 ! 0 and D44 ! 0. In this limit, the LED-based PDF Eq. (46) self-consistently reduces to pure linear advection.

3.3.1. Coefficients for the chi-squared reaction rate modelFor the covariance function in (29),

Z t

0Cjðx; x� bÞembDadb ¼ 2

XN

i¼1

/iðxÞZ t

0/iðx� bÞembDadb; ð57Þ

where m ¼ 2 or m ¼ 1 in (47) or (48), respectively. For the bases functions /iðxÞ in (30), the integral on the right-hand-sidebecomes

Fig. 8.correlat

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 333

Z t

0/kðzÞembDadb ¼ Uðb ¼ t; zÞ �Uðb ¼ 0; zÞ; ð58Þ

where

Uðb; zÞ ¼def DaL2membDa

2kkð4k2 þ Da2L2m2Þ1� cos

2kzL

� �þ 4k2

Da2L2m2þ 2k

DaLmsin

2kzL

� �" #: ð59Þ

Substituting (57)–(59) into (47) and (48) yields closed-form analytical expressions for D44 and U4, respectively. Accountingfor (23), the mean reaction rate in (48) is

hjðx;xÞi ¼XN

j¼1

/jðxÞ: ð60Þ

In contrast to their counterparts in [1], our expressions for the mean reaction rate, the diffusion coefficient D44 and the effec-tive velocity U4 vary in x. This is due to the fact that the covariance function Cjðx; yÞ we have considered is not homogeneous.

3.3.2. Coefficients for the uniform reaction rate modelSubstituting (35) into (47) and (48) yields

D44ða; tÞ ¼r2

jDa2ða� 1Þ2

2ð2Da‘� 1Þeð2Da‘�1Þt=‘ � 1 t 6 TðaÞeð2Da‘�1ÞT=‘ � 1 t > TðaÞ

(ð61Þ

and

U4ða; tÞ ¼ Dað1� aÞK � r2jDa2ða� 1Þ2ðDa‘� 1Þ

eðDa‘�1Þt=‘ � 1 t 6 TðaÞeðDa‘�1ÞT=‘ � 1 t > TðaÞ

(ð62Þ

where TðaÞ is defined in (49). The following limiting values are of interest,

D44ð0; tÞ ¼ 0; D44ð1; tÞ ¼ 0; U4ð0; tÞ ¼ KDa; U4ð1; tÞ ¼ 0: ð63Þ

Fig. 8 exhibits the dispersion coefficient D44 and effective velocity U4 for different ‘.As the correlation length increases (‘!1), D44 ! 0 and u4 ! Dað1� aÞK. In this regime, the LED-based PDF equation

(46) reduces to a first-order advection equation

@pðaÞcðx;tÞ

@t¼ �

@pðaÞcðx;tÞ

@x� DaK

@ ð1� aÞpðaÞcðx;tÞ

h i@a

: ð64Þ

This equation can also be derived directly from (21) by noting that in the limit of ‘!1 the random reaction rate (34) be-comes deterministic and equal to K. Then applying the PDF methods described in Section 2.1 directly to (21) leads to (64)without any closure approximation. This provides a demonstration of self-consistency and exactness of the LED-based clo-sure, at least in the limit of ‘!1.

3.3.3. Coefficients for the uncorrelated reaction rate modelThe exponential covariance (35) is an element of a delta sequence, i.e. it converges to r2

jdðx� yÞ as ‘! 0. This suggeststhat the coefficients for a delta-correlated random reaction rate jðx;xÞ can be obtained by taking the limit of (61), (62) as‘! 0. Since

Effective diffusivity (61) and effective velocity (62) for the exponentially correlated random reaction model. Shown are results at t ¼ 1 for differention lengths and Da ¼ 0:5;rj ¼ 1;K ¼ 2.

Fig. 9.Da ¼ 0:

334 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

lim‘!0

eðJ‘�1ÞR=‘ � 1ðJ‘� 1Þ ¼ 1 8R – 0 and 8J – 1=‘; ð65Þ

this limit yields (for a – 0)

Cjðx; yÞ ¼ r2jdðx� yÞ; D44ðaÞ ¼

Da2r2j

21� að Þ2; U4ðaÞ ¼ Da K þ Dar2

j2

� �1� að Þ: ð66Þ

It has been shown in [1] that with these coefficients the LED approximation coincides exactly with Fokker–Planck equationof the system (see also [24]). Note that if R! 0 in (65), i.e. if a! 0 (see Eqs. (61)), then the limit has to modified. A straight-forward analysis indeed shows that D44ð0Þ ¼ 0 and U4ð0Þ ¼ DaK. The plots of D44 and U4ð0Þ ¼ DaK for very small ‘ are shownin Fig. 9.

3.4. Analytical solutions for the PDF of the concentration field

In addition to solving the PDF equations introduced above, we derive a single-point PDF of the state variable cðx; t;xÞ bysolving the initial-boundary value problem (21), (22) and then using this solution to relate its PDF to the joint PDF of theinput parameters.

3.4.1. Chi-squared reaction rate modelWe show in appendix B that a solution of (21), (22) with the reaction rate jðx;xÞ parametrized by (23) has the form

cðx; t;xÞ ¼ 1� e�DaQðx;t;xÞ x P t

e�DaQðx;x;xÞ x 6 t

(; Qðx; t; xÞ ¼def XN

k¼1

vkðxÞhkðx; tÞ ð67Þ

where

hkðx; tÞ ¼def t

2kk� sinð2kx=LÞ � sin½2kðx� tÞ=L�

4kkk=L: ð68Þ

One can show that Qðx; t;xÞP 0 for all ðx; tÞ, so that the random concentration cðx; t; xÞ 2 ½0;1� with probability one. Theexponents of the product vkðxÞt in the analytical solution (67) suggests that a direct numerical simulation of the initial-boundary value problem (21), (22) using polynomial chaos or probabilistic collocation is likely to loose accuracy in time. Thisis known to yield a long-term integration problem for polynomial chaos methods [14,15]. Since Qðx; t; xÞ is a linear combi-nation of chi-squared random variables, its PDF is

pðaÞQðx;tÞ ¼1b

X1j¼0

djðxÞqNþ2jab

� �; ð69Þ

where b is an arbitrary positive parameter, qNþ2jða=bÞ is given by (27), and djðxÞ satisfy the recurrence relation (27) with /iðxÞreplaced by hiðx; tÞ. Thus, the exact one-point PDF of cðx; t;xÞ is given by ([56, p. 93])

pðaÞcðx;tÞ ¼1

Dað1� aÞ

pð� lnð1�aÞ=DaÞQðx;tÞ x P t

pð� lnð1�aÞ=DaÞQðx;xÞ x 6 t

8><>: a 2 ½0;1�: ð70Þ

This solution is analogous to the PDF solution given by Eqs. (12)–(23) in [24], which treated the random reactive rate exactly,without resorting to the expansion (23). Fig. 10 exhibits several time snapshots of the single-point PDF in (70) for Da ¼ 1 andk ¼ 1:5.

Effective diffusivity (61) and effective velocity (62) for the exponentially correlated random reaction model. Shown are results at t ¼ 1 for5;rj ¼ 1;K ¼ 2 and very small correlation lengths.

Fig. 10. Time-snapshots of the exact one-point PDF of the random concentration field (70) arising in linear reactions (a ¼ 1) for Damköhler number Da ¼ 1and the chi-squared distributed random reaction rate jðx; xÞ with k ¼ 1:5 (see §3.1).

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 335

3.4.2. Uniform reaction rate modelWe show in appendix B that a solution of (21), (22) with the reaction rate jðx;xÞ parametrized by (34) has the form

cðx; t;xÞ ¼ 1� e�DaQðx;t;xÞ x P t

e�DaQðx;x;xÞ x 6 t

(; Qðx; t; xÞ¼defKt þ

ffiffiffi3p

rjffiffiffiffiffi2‘p

XN

k¼1

fkðxÞdkðx; tÞ ð71Þ

where

dkðx; tÞ ¼def ‘

ffiffiffiffiffihkp

AksinðzkxÞ � sin½zkðx� tÞ� � cosðzkxÞ � cos½zkðx� tÞ�

‘zk

� �: ð72Þ

The procedure followed in the previous section to obtain the exact one-point PDF of the concentration field can be repeatedfor uniform random reaction rates. However, as we have anticipated in Section 3.1.2, the exact formula for the PDF of asuperimposition of uniform independent random variables is rather involved (e.g., [55,54]). Instead, we consider a non-para-metric estimate of the concentration PDF based on the analytical solution (71). Fig. 11 exhibits four time snapshots of theresulting PDF at x ¼ 25 for rj

ffiffiffi3p

=ffiffiffiffiffi2‘p

¼ 0:25;K ¼ 0:75 and different correlation lengths. The cases ‘ ¼ 300; ‘ ¼ 50 and‘ ¼ 1 are rather similar to each others. However, the random paths of the solution field cðx; t;xÞ are completely differentfor different ‘, as shown in Fig. 12.

According to the central limit theorem, for ‘! 0 the exponent in (71) is Gaussian for each fixed x and t (summation ofmany independent random variables). In addition, the variance Varðx; tÞ ¼defð1=3Þ

PNi¼1diðx; tÞ2 of the series

XN

i¼1

fiðxÞdiðx; tÞ ð73Þ

has a limit for ‘! 0 which does not depend on x (see Fig. 13). Such limit value is identified as

Varðx; tÞ !‘!02‘t=3: ð74Þ

Fig. 11. Time snapshots of the one-point PDF of the random concentration field at x ¼ 25 for rjffiffiffi3p

=ffiffiffiffiffi2‘p

¼ 0:25 and K ¼ 0:75. Shown are results obtainedwith different correlation lengths and Da ¼ 1.

Fig. 12. Realizations of the random concentration field (71) (first row) corresponding to the realizations of random reaction rate shown in the second row.The reaction rate samples are obtained for rj

ffiffiffi3p

=ffiffiffiffiffi2‘p

¼ 0:25;K ¼ 0:75 and correlation lengths ‘ ¼ 50 (left) and ‘ ¼ 1 (right). The Damköhler number is setto Da ¼ 1.

336 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

This implies that for small correlation lengths, i.e., for Gaussian white-noise random reaction rates, the PDF of the concen-tration field is Gaussian and explicitly given by

pðaÞcðx;tÞ ¼1ffiffiffiffiffiffiffi

2pp

Dað1� aÞrj

SðtÞ x P t

SðxÞ x 6 t

; SðzÞ ¼def 1ffiffiffi

zp exp �ðlnð1� aÞ=Daþ KzÞ2

2r2jz

" #: ð75Þ

Fig. 13. (a) Contour plot of Varðx; tÞ for ‘ ¼ 0:1. (b) Time evolution of Varðx; tÞ at x ¼ 100 for different correlation lengths ‘. Figure (b) shows thatVarðx; tÞ ¼ 2‘t=3 for ‘! 0 and x P t, regardless on the value of x (left).

Fig. 14. Time-snapshots of the PDF of the random concentration field at x ¼ 50 for uncorrelated (Gaussian white-noise type) random reaction rates. Shownis the comparison between the numerical solution to (52) and the analytical result (75) for K ¼ 2;Da ¼ 0:5 and rj ¼ 0:12.

Fig. 15. Time-snapshots of the PDF of the random concentration field at x ¼ 100 for weakly correlated (‘ ¼ 0:1) random reaction rates. Shown is thecomparison between the numerical solution to Eq. (52) and the PDF obtained from the analytical solution (71). We have set K ¼ 2;Da ¼ 0:5; ‘ ¼ 0:1 andrj ¼ K

ffiffiffiffiffi2‘p

=ð3ffiffiffi3pÞ in order for the random reaction rate to be positive with probability one. With these parameters, the standard deviation of the random

reaction rate is about 33% of the mean value K (see (34)).

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 337

Fig. 16. Same as Fig. 15 but with ‘ ¼ 10.

Fig. 17. Same as Fig. 15 but with ‘ ¼ 100.

338 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

4. Numerical results

In order to solve the initial/boundary value problem (52)–(56) we employed a conditionally stable time integrationscheme for advection–diffusion equations. Specifically, we discretized (52) by using a second-order central finite differencescheme with an explicit third-order Runge–Kutta time stepping. A numerical solution to (52) was then used to compute theone-point PDF by numerically differentiating PðaÞcðx;tÞ with respect to a, since

Fig. 18. Same as Fig. 15 but with ‘ ¼ 1000.

Fig. 19. Comparison between the one-point one-time PDF as obtained from the exact analytical formula (70) (continuous line) and the LED approximation(52) (dashed line). Shown are different time snapshots at location x = 100. The Damköhler number is set at Da ¼ 1, while the parameters used for therandom reaction rate (23) are N = 20 and k ¼ 1:5 (see Eq. (30)).

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 339

pðaÞcðx;tÞ ¼@PðaÞcðx;tÞ

@a: ð76Þ

The latter step may reduce the overall accuracy of the PDF representation.

340 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

4.1. Uncorrelated reaction rate model

For uncorrelated random reaction rates the LED closure is exact. Therefore, the numerical solution to (52) whose coeffi-cients are given by (66) should match exactly the analytical PDF in (75). Fig. 14 demonstrates this to be the case. A funda-mental question is whether the LED approximation produces reliable results also for weakly or strongly correlated randomreaction rates.

4.2. Uniform reaction rate model

Figs. 15–18 exhibit time snapshots of the one-point PDF of the random concentration field corresponding to random reac-tion rates with increasing correlation lengths. As the correlation length of the random reaction rate increases the LEDapproximation becomes less accurate, while continuing to maintain a good qualitative statistical description. Note that inall cases, the LED approximation leads to PDFs whose tails are fatter than those of their exact counterparts. In other words,the LED-based PDF equations provide a conservative estimate of the predictive uncertainty.

As demonstrated in Section 3.3.2, the LED approximation is exact for very large ‘. However, our positivity constraint onjðx;xÞ imposes rj ¼ K

ffiffiffiffiffi2‘p

=ðwð‘Þffiffiffi3pÞ, with wð‘Þ depending on ‘ and bounded from below (see Fig. 5). This means that, if we

set wð‘Þ ¼ 3 as we have done here we have, for ‘!1,

kðx;xÞ ! K½1þ #f1ðxÞ�; # 2 Rþ; f1ðxÞ uniform in ½�1;1�: ð77Þ

Therefore, we never reach the conditions for which the LED approximation is exact. This justifies the deviation between theanalytical solution and the numerical solution to the LED-based PDF equation observed in Fig. 18.

4.3. Chi-squared reaction rate model

For the reaction rate with the inhomogeneous correlation function introduced in Section 3.1, the diffusion coefficient D44

and effective velocity u4 in (52) are given by (47)–(59). Fig. 19 shows the time snapshots of the one-point PDF computed withboth the LED closure (52) and the exact analytical solution (70), at x ¼ 100 for k ¼ 1:5. The LED closure introduces some er-rors into the PDF dynamics. This is to be expected from the results reported in the previous section. The LED closure is exactfor white-noise Gaussian random processes, and its accuracy decreases as the correlation length increases. Despite the minoraccuracy problem, the overall advantages of the LED closure approximation are evident. In particular, the dimensionality ofthe LED approximation is much lower than that of the exact joint PDF equation. This eventually allows for an effective sim-ulation of high-dimensional advection–reaction systems.

5. Summary

We investigated two general frameworks for computing the probability density function (PDFs) of radom scalar fieldswhose dynamics is governed by advection–reaction equations (AREs) with uncertain advection velocity and reaction rates.These approaches are the single-point PDF equation based on a large-eddy diffusivity (LED) closure approximation [1], andthe joint PDF equation based on decomposition of random system parameters into finite series of random variables. We com-pared the solutions of these two equations with analytical solutions to an ARE, whose linear reaction law was parametrizedby random reaction rates defined in terms of either chi-squared or uniform random variables. This allowed us to test theaccuracy of the LED-based PDF equation on a reliable basis. We found that the LED closure is exact both for white-noiseGaussian random reaction rates and for reaction rates having an infinite correlation length. For intermediate correlationlengths, the accuracy of the LED closure decreases, while continuing to maintain a good qualitative statistical description.In all cases, the LED approximation leads to PDFs, whose tails are fatter than those of their exact counterparts. In other words,the LED-based PDF equations provide a conservative estimate of the predictive uncertainty.

The computational advantages of the closure approximation are evident for input uncertainties characterized in terms ofa large number of random variables. Indeed, the exact PDF (or CDF) equation of the system in these cases is high-dimensionaland therefore its solution requires appropriate numerical techniques such as sparse grid [57,31,7] or functional ANOVA[58,32,33]. More generally, for stochastic PDEs of order greater than one closures are unavoidable since an exact exact equa-tion for the one-point one-time PDF does not exist due to the non-locality (in space and time) of the solution. In these cases,the closure may be based, e.g., on a truncation of the exact Hopf functional equation leading to a hierarchy of equations forthe cumulants of the solution [59,18,17] or to a truncated Lundgren-Monin-Novikov hierarchy [21]. Alternatively other clo-sures such as direct interaction approximations [43,60], conditional closures [61,62] and mapping closures [63,64] (see also[65]) can be considered.

Acknowledgments

We acknowledge financial support from DOE (Grant No. DE-FG02-07ER25818), OSD-MURI (Grant No. FA9550-09-1-0613)and NSF (Grant No. DMS-0915077).

D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 341

Appendix A. Analytical solution to the exact joint PDF equation

We solve the initial-boundary value problem (40)–(45) by using the method of characteristics [47]. The (random) char-acteristics curves of the system, tðs; t; x; aÞ; xðs; t; x; aÞ; aðs; t; x; aÞ and zðs; t; x; bÞ, satisfy the equations

@t@s¼ 1;

@x@s¼ 1;

@a@s¼ DaHðxÞð1� aÞ; @z

@s¼ DaHðxÞz; HðxÞ¼defXN

i¼1

/iðxÞbi; ðA:1Þ

subject to one of the three sets of auxiliary conditions

tð0; t; x; aÞ ¼ 0; xð0; t; x; aÞ ¼ x; að0; t; x; aÞ ¼ a; zð0; t; x; aÞ ¼ dðaÞpðbÞv ; ðA:2Þ

tð0; t; x; aÞ ¼ t; xð0; t; x; aÞ ¼ 0; að0; t; x; aÞ ¼ a; zð0; t; x; aÞ ¼ dðaÞpðbÞv ; ðA:3Þ

or

tð0; t; x; aÞ ¼ t; xð0; t; x; aÞ ¼ x; að0; t; x; aÞ ¼ 0; zð0; t; x; aÞ ¼ 0; ðA:4Þ

depending on whether the characteristics originate at the ða; x; 0Þ; ða;0; tÞ, or ð0; x; tÞ plane, respectively.Let us consider the set of characteristic curves departing from the ða; x;0Þ plane (the analogous procedure is to be per-

formed for the other two cases). Integration of the system (A.1) with the initial conditions (A.2) yields

t ¼ s; x ¼ sþ x;Z a

að0Þ

db1� b

¼ DaIðs; x;bÞ;Z z

zð0Þ

dbb¼ DaIðs; x;bÞ: ðA:5Þ

where

Iðs; x;bÞ¼defZ s

0Hðxþ sÞds ¼

XN

k¼1

bkðxÞkk

s2� sinð2kðxþ sÞ=LÞ � sinð2kx=LÞ

4k=L

� �: ðA:6Þ

Integrating the last equation in (A.5) we obtain

z ¼ dðaÞpðbÞv eDa Iðs;x;bÞ: ðA:7Þ

Solving the third equation in (A.5) for a ¼ að0Þ, and substituting the result into (A.7), gives

pða;bÞcðx;tÞv ¼ d aþ e�DaIðt;x�t;bÞ� �pðbÞv eDaIðt;x�t;bÞ: ðA:8Þ

Note that this representation involves a nonlinear transformation inside the delta function, which needs to be properly han-dled [42].

Repeating these mathematical steps for the characteristic curves arising from the other two planes, ða;0; tÞ and ð0; x; tÞ, weobtain the analytical expression for the joint PDF pða;bÞcðx;tÞv that is valid over its domain of definition. The regions of the spaceða; x; tÞ where each of the three families of characteristics holds is determined by studying the characteristic curves arisingfrom the intersections of the planes ða; x;0Þ; ða;0; tÞ and ð0; x; tÞ, namely, the lines ða;0;0Þ; ð0; x;0Þ and ð0;0; tÞ. Finally, an inte-gration of the analytical expression for pða;bÞcðx;tÞv with respect to b1; . . . ; bN yields the one-point PDF of the concentration field(70).

Appendix B. Analytical solution to the advection–reaction problem in physical space

We solve the initial-boundary value problem (21), (22) by using the method of characteristics [47]. The (random) char-acteristic curves of the system, tðs; t; xÞ; xðs; t; xÞ, zðs; t; x;xÞ, satisfy the equations

@t@s¼ 1;

@x@s¼ 1;

@z@s¼ ð1� zÞjðx;xÞDa; ðB:1Þ

subject to the initial conditions

tð0; t; xÞ ¼ 0; xð0; t; xÞ ¼ x; zð0; t; x;xÞ ¼ 0; ðB:2Þ

or

tð0; t; xÞ ¼ t; xð0; t; xÞ ¼ 0; zð0; t; x;xÞ ¼ 0; ðB:3Þ

depending on where the characteristic comes from, i.e., from the ðx; 0Þ axis or the ð0; tÞ axis, respectively. Let us first considerthe characteristics arising from the ðx;0Þ axis. By integrating (B.1) with initial conditions (B.2) we obtain

t ¼ s; x ¼ xþ s;Z zðsÞ

0

db1� b

¼ DaZ s

0jðxþ sÞds: ðB:4Þ

342 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343

Depending on the specific choice of the random reaction rate jðx;xÞ (see section 3.1.1 and Section 3.1.2) the integral appear-ing in the last equation in (B.4) produces different expressions. For example, by using the representation (23) with /k givenin (30), we obtain

zðs; x;xÞ ¼ 1� exp �DaXN

k¼1

vkðxÞkk

s2� sinð2kðxþ sÞ=LÞ � sinð2kx=LÞ

4k=L

� �" #: ðB:5Þ

Similarly, along the characteristics curves arising from the ð0; tÞ axis (i.e., t ¼ t þ s and x ¼ s) we obtain

zðs; x;xÞ ¼ 1� exp �DaXN

k¼1

vkðxÞkk

s2� sinð2ks=LÞ

4k=L

� �" #: ðB:6Þ

Expressing x and s in terms of the corresponding values t and x along the two groups of characteristics gives the analyticalsolution (67). Repeating these mathematical steps for the reaction rate model (34) gives the analytical solution (71).

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